Quantitative Endpoint Transport for the Collatz Map with a Natural-Density Descent Application

Version 2.2.0 isolates and strengthens a reusable arbitrary-target endpoint-transport theorem for the shortcut Collatz map. A power-saving exceptional-set estimate pulls back through one floor(log_2 n)-step endpoint with exponent asymptotically linear in the input density exponent. The refined prefactor is sublinear in the incoming density constant and has the inverse-density power stated in Theorem 5.3. Exact parity coordinates and fixed-total compositions reduce endpoint multiplicities to a Rényi estimate, and Hölder's inequality combines that estimate with arbitrary target mass. An endpoint-only bootstrap then gives natural-density descent to exp((log n)^(1-delta)) for every delta below 0.251245530155874..., together with a stretched-exponential exceptional count and explicit logarithmic witness time. The manuscript proof is self-contained. A focused Lean 4.33 formalization of the refined arbitrary-target operator is archived separately at https://doi.org/10.5281/zenodo.22544010 and tagged v2.2.0 at the public repository. These are almost-all results. They do not prove the Collatz conjecture or exclude exceptional cycles or divergent trajectories.